Plethystic formulas for permutation enumeration

Abstract

We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent number, peak number, left peak number, and the number of up-down runs. We apply these results to cyclic permutations, involutions, and derangements, and more generally, to derive formulas for counting all permutations by the above statistics jointly with the number of fixed points and jointly with cycle type. A number of known formulas are recovered as special cases of our results, including formulas of Désarménien–Foata, Gessel–Reutenauer, Stembridge, Fulman, Petersen, Diaconis–Fulman–Holmes, Zhuang, and Athanasiadis.